#6828227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,31)(21,23)(22,26)(25,33)(27,39)(28,36)(29,35)(30,44)(32,40)(34,45)(37,41)(38,43)(42,47)(46,48);(1,3,6,2)(4,8,5,7)(18,23)(19,27)(20,32)(22,28)(24,37)(25,40)(29,41)(31,45)(33,34)(35,39)(38,44)(42,46);(10,13)(14,16)(17,19,28,41,21,29,22,27)(18,24,36,37,23,35,26,39)(20,30,32,46,33,43,34,42)(25,38,45,47,31,44,40,48);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,23)(19,29)(25,31)(26,36)(27,41)(30,43)(40,45)(42,46);(1,5,6,4)(2,7,3,8)(17,20,28,32,21,33,22,34)(18,25,36,45,23,31,26,40)(19,30,41,46,29,43,27,42)(24,38,37,47,35,44,39,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=fgi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6735423);